direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C3×C42.C2, C42.4C6, C12.11Q8, C4⋊C4.4C6, C4.3(C3×Q8), C2.4(C6×Q8), C6.21(C2×Q8), (C4×C12).10C2, C6.45(C4○D4), (C2×C6).80C23, (C2×C12).67C22, C22.15(C22×C6), (C2×C4).7(C2×C6), C2.8(C3×C4○D4), (C3×C4⋊C4).11C2, SmallGroup(96,172)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C42.C2
G = < a,b,c,d | a3=b4=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=bc2, dcd-1=b2c >
Subgroups: 68 in 56 conjugacy classes, 44 normal (12 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C6, C2×C4, C2×C4, C12, C12, C2×C6, C42, C4⋊C4, C2×C12, C2×C12, C42.C2, C4×C12, C3×C4⋊C4, C3×C42.C2
Quotients: C1, C2, C3, C22, C6, Q8, C23, C2×C6, C2×Q8, C4○D4, C3×Q8, C22×C6, C42.C2, C6×Q8, C3×C4○D4, C3×C42.C2
►Regular action on 96 points
Generators in S96
(1 35 11)(2 36 12)(3 33 9)(4 34 10)(5 30 14)(6 31 15)(7 32 16)(8 29 13)(17 37 41)(18 38 42)(19 39 43)(20 40 44)(21 51 45)(22 52 46)(23 49 47)(24 50 48)(25 95 69)(26 96 70)(27 93 71)(28 94 72)(53 75 79)(54 76 80)(55 73 77)(56 74 78)(57 61 83)(58 62 84)(59 63 81)(60 64 82)(65 87 91)(66 88 92)(67 85 89)(68 86 90)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)(25 26 27 28)(29 30 31 32)(33 34 35 36)(37 38 39 40)(41 42 43 44)(45 46 47 48)(49 50 51 52)(53 54 55 56)(57 58 59 60)(61 62 63 64)(65 66 67 68)(69 70 71 72)(73 74 75 76)(77 78 79 80)(81 82 83 84)(85 86 87 88)(89 90 91 92)(93 94 95 96)
(1 23 39 15)(2 24 40 16)(3 21 37 13)(4 22 38 14)(5 34 52 42)(6 35 49 43)(7 36 50 44)(8 33 51 41)(9 45 17 29)(10 46 18 30)(11 47 19 31)(12 48 20 32)(25 89 64 80)(26 90 61 77)(27 91 62 78)(28 92 63 79)(53 94 66 81)(54 95 67 82)(55 96 68 83)(56 93 65 84)(57 73 70 86)(58 74 71 87)(59 75 72 88)(60 76 69 85)
(1 74 39 87)(2 88 40 75)(3 76 37 85)(4 86 38 73)(5 28 52 63)(6 64 49 25)(7 26 50 61)(8 62 51 27)(9 54 17 67)(10 68 18 55)(11 56 19 65)(12 66 20 53)(13 58 21 71)(14 72 22 59)(15 60 23 69)(16 70 24 57)(29 84 45 93)(30 94 46 81)(31 82 47 95)(32 96 48 83)(33 80 41 89)(34 90 42 77)(35 78 43 91)(36 92 44 79)
G:=sub<Sym(96)| (1,35,11)(2,36,12)(3,33,9)(4,34,10)(5,30,14)(6,31,15)(7,32,16)(8,29,13)(17,37,41)(18,38,42)(19,39,43)(20,40,44)(21,51,45)(22,52,46)(23,49,47)(24,50,48)(25,95,69)(26,96,70)(27,93,71)(28,94,72)(53,75,79)(54,76,80)(55,73,77)(56,74,78)(57,61,83)(58,62,84)(59,63,81)(60,64,82)(65,87,91)(66,88,92)(67,85,89)(68,86,90), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48)(49,50,51,52)(53,54,55,56)(57,58,59,60)(61,62,63,64)(65,66,67,68)(69,70,71,72)(73,74,75,76)(77,78,79,80)(81,82,83,84)(85,86,87,88)(89,90,91,92)(93,94,95,96), (1,23,39,15)(2,24,40,16)(3,21,37,13)(4,22,38,14)(5,34,52,42)(6,35,49,43)(7,36,50,44)(8,33,51,41)(9,45,17,29)(10,46,18,30)(11,47,19,31)(12,48,20,32)(25,89,64,80)(26,90,61,77)(27,91,62,78)(28,92,63,79)(53,94,66,81)(54,95,67,82)(55,96,68,83)(56,93,65,84)(57,73,70,86)(58,74,71,87)(59,75,72,88)(60,76,69,85), (1,74,39,87)(2,88,40,75)(3,76,37,85)(4,86,38,73)(5,28,52,63)(6,64,49,25)(7,26,50,61)(8,62,51,27)(9,54,17,67)(10,68,18,55)(11,56,19,65)(12,66,20,53)(13,58,21,71)(14,72,22,59)(15,60,23,69)(16,70,24,57)(29,84,45,93)(30,94,46,81)(31,82,47,95)(32,96,48,83)(33,80,41,89)(34,90,42,77)(35,78,43,91)(36,92,44,79)>;
G:=Group( (1,35,11)(2,36,12)(3,33,9)(4,34,10)(5,30,14)(6,31,15)(7,32,16)(8,29,13)(17,37,41)(18,38,42)(19,39,43)(20,40,44)(21,51,45)(22,52,46)(23,49,47)(24,50,48)(25,95,69)(26,96,70)(27,93,71)(28,94,72)(53,75,79)(54,76,80)(55,73,77)(56,74,78)(57,61,83)(58,62,84)(59,63,81)(60,64,82)(65,87,91)(66,88,92)(67,85,89)(68,86,90), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48)(49,50,51,52)(53,54,55,56)(57,58,59,60)(61,62,63,64)(65,66,67,68)(69,70,71,72)(73,74,75,76)(77,78,79,80)(81,82,83,84)(85,86,87,88)(89,90,91,92)(93,94,95,96), (1,23,39,15)(2,24,40,16)(3,21,37,13)(4,22,38,14)(5,34,52,42)(6,35,49,43)(7,36,50,44)(8,33,51,41)(9,45,17,29)(10,46,18,30)(11,47,19,31)(12,48,20,32)(25,89,64,80)(26,90,61,77)(27,91,62,78)(28,92,63,79)(53,94,66,81)(54,95,67,82)(55,96,68,83)(56,93,65,84)(57,73,70,86)(58,74,71,87)(59,75,72,88)(60,76,69,85), (1,74,39,87)(2,88,40,75)(3,76,37,85)(4,86,38,73)(5,28,52,63)(6,64,49,25)(7,26,50,61)(8,62,51,27)(9,54,17,67)(10,68,18,55)(11,56,19,65)(12,66,20,53)(13,58,21,71)(14,72,22,59)(15,60,23,69)(16,70,24,57)(29,84,45,93)(30,94,46,81)(31,82,47,95)(32,96,48,83)(33,80,41,89)(34,90,42,77)(35,78,43,91)(36,92,44,79) );
G=PermutationGroup([[(1,35,11),(2,36,12),(3,33,9),(4,34,10),(5,30,14),(6,31,15),(7,32,16),(8,29,13),(17,37,41),(18,38,42),(19,39,43),(20,40,44),(21,51,45),(22,52,46),(23,49,47),(24,50,48),(25,95,69),(26,96,70),(27,93,71),(28,94,72),(53,75,79),(54,76,80),(55,73,77),(56,74,78),(57,61,83),(58,62,84),(59,63,81),(60,64,82),(65,87,91),(66,88,92),(67,85,89),(68,86,90)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24),(25,26,27,28),(29,30,31,32),(33,34,35,36),(37,38,39,40),(41,42,43,44),(45,46,47,48),(49,50,51,52),(53,54,55,56),(57,58,59,60),(61,62,63,64),(65,66,67,68),(69,70,71,72),(73,74,75,76),(77,78,79,80),(81,82,83,84),(85,86,87,88),(89,90,91,92),(93,94,95,96)], [(1,23,39,15),(2,24,40,16),(3,21,37,13),(4,22,38,14),(5,34,52,42),(6,35,49,43),(7,36,50,44),(8,33,51,41),(9,45,17,29),(10,46,18,30),(11,47,19,31),(12,48,20,32),(25,89,64,80),(26,90,61,77),(27,91,62,78),(28,92,63,79),(53,94,66,81),(54,95,67,82),(55,96,68,83),(56,93,65,84),(57,73,70,86),(58,74,71,87),(59,75,72,88),(60,76,69,85)], [(1,74,39,87),(2,88,40,75),(3,76,37,85),(4,86,38,73),(5,28,52,63),(6,64,49,25),(7,26,50,61),(8,62,51,27),(9,54,17,67),(10,68,18,55),(11,56,19,65),(12,66,20,53),(13,58,21,71),(14,72,22,59),(15,60,23,69),(16,70,24,57),(29,84,45,93),(30,94,46,81),(31,82,47,95),(32,96,48,83),(33,80,41,89),(34,90,42,77),(35,78,43,91),(36,92,44,79)]])
C3×C42.C2 is a maximal subgroup of
C42.8D6 Dic6.4Q8 C42.68D6 C42.215D6 D12.4Q8 C42.70D6 C42.216D6 C42.71D6 Dic6⋊7Q8 C42.147D6 C42.236D6 C42.148D6 D12⋊7Q8 C42.237D6 C42.150D6 C42.151D6 C42.152D6 C42.153D6 C42.154D6 C42.155D6 C42.156D6 C42.157D6 C42.158D6
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 6A | ··· | 6F | 12A | ··· | 12L | 12M | ··· | 12T |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | ||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | Q8 | C4○D4 | C3×Q8 | C3×C4○D4 |
| kernel | C3×C42.C2 | C4×C12 | C3×C4⋊C4 | C42.C2 | C42 | C4⋊C4 | C12 | C6 | C4 | C2 |
| # reps | 1 | 1 | 6 | 2 | 2 | 12 | 2 | 4 | 4 | 8 |
Matrix representation of C3×C42.C2 ►in GL4(𝔽13) generated by
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 12 | 0 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 12 | 0 |
| 11 | 9 | 0 | 0 |
| 4 | 2 | 0 | 0 |
| 0 | 0 | 3 | 9 |
| 0 | 0 | 9 | 10 |
G:=sub<GL(4,GF(13))| [9,0,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[8,0,0,0,0,8,0,0,0,0,0,12,0,0,1,0],[0,1,0,0,1,0,0,0,0,0,0,12,0,0,1,0],[11,4,0,0,9,2,0,0,0,0,3,9,0,0,9,10] >;
C3×C42.C2 in GAP, Magma, Sage, TeX
C_3\times C_4^2.C_2
% in TeX
G:=Group("C3xC4^2.C2"); // GroupNames label
G:=SmallGroup(96,172);
// by ID
G=gap.SmallGroup(96,172);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-2,-2,144,313,295,938,122]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^4=c^4=1,d^2=c^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b*c^2,d*c*d^-1=b^2*c>;
// generators/relations